One Star Quests
A Dangerous Cocktail
  • Unlocked: Third day
  • Quest item: Clear Ooze x 4
  • Reward: 200 gil, Caution Beacon

Clear Oozes are dropped by Flanitors, which appear often in the streets of Yusnaan alongside Personal Guards. Area attacks will make killing them easy.
A Man for a Chocobo Girl
  • Unlocked: Complete 2-1 of the main quest in Yusnaan
  • Quest Item: Chocobo Girl's Phone Number
  • Reward: 400 gil, Afro & Red Chick, Afro & Blue Chick

The Chocobo Girl's Phone Number can be found on a poster in a back alleyway in the Reveler's Quarter. Look for the two Diviners and you'll find the alley nearby.
Secret Machine
  • Unlocked: First visit
  • Quest Item: Ether Coil x 3
  • Reward: 150 gil, Silver Padlock

Ether Coils are dropped by Desdemonas. They can be found aplenty around the Palace of the Patron.
Source of Inspiration
  • Unlocked: Fifth day
  • Quest Item: Shattered Bone x 3
  • Reward: 180 gil, Water Spirit Wings

Shattered Bones are dropped by Skata'ne, the flailing bird-like creatures found in the Lower City. They're a little uncommon, and more often you'll run into more dangerous creatures, so be careful down there.
Two Star Quests
Beast Summoner
  • Unlocked: Seventh Day
  • Quest Item: Shattered Bone x 5, Poisonous Sting x 10
  • Reward: 600 gil, Single Horn, Antler

Shattered Bones are gathered from Skatan'es, while Poisonous Stings come from Anubyses. Skatan'es can be found in South City, while Anubyses pop up more often in the Warehouse District. You can find both in the Patron's Palace.
Rebuilding
  • Unlocked: After toppling God's statue in the Augur's Quarter
  • Quest Item: Statue Fragment x 5
  • Reward: 500 gil, Bushy Mustache

Hunt around the Augur's Quarter, the Reveler's Quarter, and the front of the Patron's Palace for Statue Fragments. They appear as gleaming items on the ground.
Soulful Horn
  • Unlocked: First day
  • Quest Item: Poisonous Sting x 10
  • Reward: 400 gil, Bubbly Party Hat, Rum Raisin Ice Cream

Poisonous Stings are dropped by Anubys, the first enemy you ever fought in this iteration of Final Fantasy XIII. They pop up occasionally in the streets of Luxerion, typically in open spaces like the squares in front of the train stations.
Time Doesn't Heal
  • Unlocked: Kill the Cyclops in the Warehouse District
  • Quest Item: Jade Hair Comb, Bronze Pocket Watch
  • Reward: 800 gil, Sentimental Parasol, Mark of Lindzei

Both items can be found in the same general area, though you'll probably pick them up at different times. The Bronze Pocket Watch can be found along the 'secret' path in Yusnaan during your first trip through; search a side catwalk as you're approaching the midway point of this area to find it. The Jade Hair Comb is located right at the end of this area, but you'll have to come back after beating the Cyclops and essentially ruining the balcony view to find it.
True Colors
  • Unlocked: Fifth day
  • Quest Item: Firewyrm Scale
  • Reward: 600 gil, Lebreau's Black Tattoo, Lebreau's Green Tattoo

Firewyrm Scales are dropped by Zaltys, the first boss in the game. I presume you killed one? Then you should have the Firewyrm Scale already. Handy! If you've used it up in another way, you can fight additional Zaltys (Zalti?) in the Patron's Palace.
Unfired Firework
  • Unlocked: After completing the main quests of Yusnaan
  • Quest Item: Unfired Rocket Fireworks
  • Reward: 900 gil, Twilight Shades, Stormy Motors Logo

After the events of Yusnaan's main quests, there will be four fireworks stations left in the streets. Collect three Fireworks (the Chocobo Girls will offer you one, and there are plenty to gather in excess of ten during the chapter) and find one of these stations. Test them until one of the fireworks fails to go off.
Youth Potion
  • Unlocked: Seventh Day
  • Quest Item: Single Eye
  • Reward: 700 gil, Lucky Clover, Festive Tree

Single Eyes are dropped by, duh, Cyclopses. You'll get one by killing the Cyclops that attacks you in the main quest of Yusnaan.
Three Star Quests
Spell for Spell
  • Unlocked: Seventh Day
  • Quest Item: Radial Bearing x 30
  • Reward: 1300 gil, Light Bulb, Blue Guitar

Radial Bearings are acquired by destroying Hoplites. They're found in the warehouse of the Lower City where you go on the Secret Tour.
Ultimate Craving
  • Unlocked: After completing Secret Machine, Spell for Spell, and A Dangerous Cocktail
  • Quest Item: Radial Bearing x 10, Ether Coil x 15, Clear Ooze x 30
  • Reward: 1600 gil, Wind-Up Halo, Red Propellor, Ether

Radial Bearings are dropped by Hoplites, found in the Industrial District. Ether Coils are dropped by Desdemonas, which appear past the halfway point of the same area. Clear Oozes are dropped by Flanitors, found throughout the streets of Yusnaan.
What Seekers Seek
  • Unlocked: Seventh Day, complete Youth Potion and True Colors
  • Quest Item: Firewyrm Scale x 5, Single Eye x 5
  • Reward: 1200 gil, Prophetic Headdress, Saint's Beard

Firewyrm Scales are dropped by Zaltys, while Single Eyes are dropped by Cyclopses. Zaltyses appear in the tunnels of the Patron's Palace, while Cyclopses appear in the Ballroom of that same Palace.